Logarithmic Time Parallel Algorithms for Recognizing Comparability and Interval Graphs

We give fast parallel algorithms for recognizing ad representing comparability graphs that can be transitively oriented, and interval graphs, the intersection graphs of intervals along the real line. Under the CRCW PRAM model, both algorithms use $O(n^{3})$ processors in $O$(log $n$) time to check if a graph belongs to the desired class, and if it does then a valid representation of the graph can be produced. The algorithms gain their efficiency by using fast algorithms for finding the modular decomposition of a graph. Both problems were known to be in $NC$, but the known algorithms require more time than ours does